# Total energy & mass in different reference frames

Discussion in 'Physics & Math' started by Speakpigeon, Jun 22, 2018.

1. ### SpeakpigeonRegistered Senior Member

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If we compare the views from two different reference frames, in motion relatively to each other, at a constant velocity close to the speed of light, and one of them broadly at rest with respect to the entire universe, and the other one associated with an object having a small mass at rest. What are the differences in terms of total mass and total energy of the universe?
I would assume the overall rest masses to be identical. But from the frame broadly at rest, there's just the one small object that's near the speed of light. The energy involved is just the energy required to speed up the small object. No problem.
From the point of view of the frame associated with the small object, however, it's the rest of the universe which is speeding up and with a velocity close to c. Where could the energy necessary to accelerate the entire universe possibly come from? And if we dismiss that perspective and say there's no energy involved except the one necessary to accelerate the small object, then we also have to dismiss the relativistic mass of the rest of the universe seen as speeding at near the speed of light relatively to the frame of the small object.
This seems to suggest that relativistic mass is more a fiction than a reality.
Or where do I have it wrong?
EB

3. ### Q-reeusValued Senior Member

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In modern cosmology there is no such thing as a center of the universe. Everywhere looks (almost) like the center. Best that can be done is to say one is at rest wrt the local Hubble flow. Meaning the CMBR has no dipole anisotropy. Our solar system has a drift of ~ 370 km/s wrt one with a uniform CMBR:
http://www.astro.ucla.edu/~wright/CMB-DT.html
But just for sake of argument suppose there was a meaningful center of the universe. To logically distinguish between claiming the 'rest of the universe' has accelerated to high velocity, as opposed to a small mass, say you in a spaceship, it's only necessary to ask which entity experienced g-forces in changing relative speed. Clearly that will be you in the spaceship, not 'the rest of the universe'. Further, what caused the acceleration? Your spaceship firing up to produce an exhaust out the back. Similarly for any other means to add momentum to an object e.g. charged particle accelerator.
So there is no real ambiguity or paradox once all factors are considered.

5. ### SpeakpigeonRegistered Senior Member

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I wasn't assuming a "center of the universe", only that we can meaningfully assign frames of reference to simple things like a small moving object and the rest of the universe. The term "broadly" could even be conceived as referring to a fictitious universe limited to our galaxy for example. I guess any really big object would do.
The question also wasn't about which bit of matter will have been accelerated and which won't. It's clear in my question which is which.
However, the small object stops being accelerated at some point and the question is about the comparison between two inertial frames of reference. As I understand it, both are legitimate frames for measuring the characteristics of physical things.
EB

Last edited: Jun 22, 2018

7. ### Q-reeusValued Senior Member

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Yes the heart and sole of SR is that there is no preferred frame of reference. In cosmological setting, the CMBR sort of changes that, but only in the sense of 'looking out the window', not in terms of local physics.

8. ### SpeakpigeonRegistered Senior Member

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Your answer seems to me to suggest that an observer at rest in the reference frame of the small object has to dismiss the relativistic mass of the rest of the universe seen as speeding at near the speed of light. The measure of relativist mass in the reference frame of the small object is nonsensical? So, broadly, all not-at-rest mass is relative, rather than, say, substantial.
EB

9. ### Q-reeusValued Senior Member

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I'm responding to a philosopher. Bad idea maybe. Words can mean different things to different people. Not sure who first said it, but basically, the difference between philosophy and physics is the latter has to conform to experimental or observational confirmation. Only what can be measured is supposed to count. You may question that when it comes to say string/M theory which is considered by advocates as 'real physics'. But most 'practical' physicists likely disagree and call it metaphysics. What's in a name? A lot. Depending.

Getting back on track re OP specifics, one consequence of moving at really near light speed wrt CMBR i.e. 'center of universe' is that owing to the real physical consequences of SR, you would be burned to a crisp. Because of relativistic Doppler shift of CMBR frequencies. Shift from microwave to gamma ray frequencies can be nasty.
If you want to avoid that, choose a universe with zero CMBR. Might be hard to find though in practice.

10. ### Janus58Valued Senior Member

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Let's consider an equivalent example using just Newtonian Physics. You are floating in space, some distance from a object with the same mass as the Moon and at rest with respect to it. You accelerate to 5 m/s. If your mass is 70 kg, your kinetic energy as measured by someone still at rest with respect to the object will now be 875 joules. However, in your frame,it is the object that has a velocity of 5 m/sec relative to you, and with a mass of 7.35e22 kg, has a kinetic energy of 9.19e23 joules of energy after you accelerated, compared to zero before you accelerated. So is KE a fiction?

That being said, the term "relativistic mass" has pretty much been abandoned in modern physics, partly due to the kind of confusion it can cause. Instead, it is treated as an an part of the object's energy.

In Newtonian physics KE is found by
$KE = \frac{mv^2}{2}$

With Relativity it is

$KE = mc^2 \left ( \frac{1}{\sqrt{1-\frac{v^2}{c^2}}} -1 \right )$

The difference being that the Newtonian expression increases to infinity as v approaches infinity, and the Relativistic expression increases to infinity as v approaches c.

But in both cases, the KE is frame dependent.

11. ### SpeakpigeonRegistered Senior Member

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I'm no philosopher, sorry. I like to think of myself as a rational being but I certainly don't qualify as a philosopher.
I'm discussing not some philosophical perspective on the physical world but what I understand of Relativity.
This is interesting but it seems really beside the point.
EB

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265
Exactly.
EB

13. ### James RJust this guy, you know?Staff Member

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Basically, the answer to this is: it comes from nowhere.

The frame of an accelerating object is a non-inertial reference frame. In any such frame, the "usual" laws of physics don't hold, including the law of energy conservation. That's why we try to do physics in inertial frames.

14. ### Q-reeusValued Senior Member

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???? Which is saying all that apparent energy increase - a Lorentz boosted 'universe' - is got for free! Instead, asymmetry of proper accelerations pins down which entity experiences a gain in energy relative to the greater whole. And that is assuming both small object and 'rest of universe' were strictly at rest wrt each other before any such boost.
Since the perspective has been acceleration of non-gravitational origin, conservation of energy holds strictly for the entire system involved. Or you can quote a reputable source that agrees with your claim there?
It's often simpler to do there, but working in accelerated frames does not lead to weird, law-breaking physics. Maxwell's equations for instance will look different, but still obey the basics.

15. ### James RJust this guy, you know?Staff Member

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31,255
Q-reeus:

Yep. If you're sitting on an accelerating rocket, the acceleration of the universe that you observe has literally no cause. It comes for free, in that frame of reference. If you like, you can fiddle with things by introducing so-called "inertial" or "imaginary" forces to act on all objects that you observe, and say that those imaginary forces cause the kinetic energy increase that you measure. But those forces are uncaused, as far as you can tell, unless you look outside your own frame of reference.

All you're saying here is that it is possible to identify which reference frame is inertial and which is not. I agree with you.

Only in an inertial reference frame, like I said.

Any competent introductory text on relativity (special or general) will agree with me.

Yes, it does. There was a reason that one of Einstein's two postulates of special relativity was that the laws of physics are the same in all inertial frames. They simply do not work in non-inertial frames, without the introduction of imaginary forces such as the ones I have described above.

Maxwell's equations were, of course, one of the main motivators for Einstein to invent special relativity. They are not invariant in non-inertial frames, and they are only invariant in inertial frames under the assumptions of Lorentz invariance (which follows, of course, from the same basic postulates of relativity).

16. ### Q-reeusValued Senior Member

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That's not properly answering Speakpigeon's question you quoted in #10:
"...Where could the energy necessary to accelerate the entire universe possibly come from?"
The proper answer surely is there is no such gain. It's a misinterpretation of what SR implies wrt reciprocity of inertial frames to assume there could be such a real gain.
Can't follow that. An on-board accelerometer records real g-forces if you not the universe has changed relative speed.
For a linearly accelerated frame, even without a total accounting including source of acceleration, equivalence principle says everything within that frame acts as for a uniform gravitational field. Which is a conservative field and any internal physics conducted there will comply with that.
Only if a circularly accelerated frame is considered, will there be apparent violations of energy conservation within that frame, owing to Coriolis forces. But no violation occurs for the system as a whole.
[Actually, I'll take even that above bit back. Coriolis forces act always normal to any velocities, and so are conservative within that accelerated frame. Similar to how it is with action of magnetic fields on moving charges in EM. In order for energy conservation to fail, there must be non-conservative fields acting over a cyclic process. That doesn't occur in uniformly accelerated frames. One would have to go to non-uniform linearly accelerated or non-uniformly circular accelerated frames to have apparent internal violations. Which is a far more restrictive situation than you stated.]
When the laws are covariantly formulated, they hold in any frame whatsoever. Same comment re your take on ME's later.

Last edited: Jun 23, 2018
17. ### James RJust this guy, you know?Staff Member

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Q-reeus:

Consider two objects in space, A and B, initially at rest relative to one another. Suppose A accelerates towards B. In the frame of A, B is observed to accelerate from rest. If B's kinetic energy at any time is $(1/2)mv^2$, then as B is observed to accelerate, B's kinetic energy increases over time. Where does that energy come from?

18. ### SpeakpigeonRegistered Senior Member

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I considered only inertial frames of reference by considering only the moments before and after the acceleration of the small object. As seen from this vantage point, the energy of the whole universe is not conservative, unless we decide to see relativist mass and energy as unsubstantial. Either way, matter isn't an invariant of the universe. Only mass at rest may be. If we have to go back to argue from the fact that only the small object is subjected to "real" acceleration, then we also end up dismissing the relativist mass and energy of the whole universe as unsubstantial compared to the "real" mass and energy of the small object subjected to "real" acceleration.
EB

19. ### James RJust this guy, you know?Staff Member

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But in that case, you're measuring the energy in two different inertial frames - one the frame of the small object before acceleration, and the other the (different) frame of the small object after the acceleration. Why would you expect the kinetic energy measured in two different frames to be the same?

Acceleration isn't relative, though. A person who is accelerating can tell that he is accelerating, compared to a person who is not.

20. ### Q-reeusValued Senior Member

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Yes it's possible to have a pov that it's B that has gained KE without doing anything in it's inertial frame. The perspective given 1st para in #11 is imo the proper one to sort such a situation out. Sure even if both A & B have equal and opposite co-linear proper accelerations, from some 3rd inertial frame, where initial velocities are not equal and opposite, energy gains will not be equal after accelerations cease. Such frame dependent arbitrariness is of course a fundamental aspect of SR. Which is why invariants such as 4-velocity and 4-momentum are often preferred. While energy, 3-velocity, 3-momentum etc. are entirely frame dependent, by considering all aspects, it's possible in most cases to meaningfully assign energy gain/loss to a given entity. Like a bullet fired from a rifle. I think we agree on that.

21. ### SpeakpigeonRegistered Senior Member

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Sure but this means the energy and mass of the whole universe as measured by an inertial reference frame is relative to it. Each frame comes up with a different answer. The energy and mass of the whole universe as measured by an inertial reference frame is therefore "unsubstantial", i.e. it's not an invariant.
Yes, but, sorry to repeat myself, this means the energy and mass of the whole universe as measured by an inertial reference frame is relative to it and is therefore "unsubstantial".
And the relativist increase of the energy or mass of the small object is similarly unsubstantial, even though it is really accelerated. It seems to me to behaves more like potential energy in a gravitational field.
EB

22. ### James RJust this guy, you know?Staff Member

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Speakpigeon:

Essentially, yes, but it's a bit tricky. How are you going to measure "the energy" of the whole universe, exactly?

Another thing to mention is that if you're talking about rest mass, then that is an invariant quantity. Everybody, in whatever frame, agrees on that.

But the basic point, applied to my example with a "universe" of just two objects, A and B, above, is correct. The energy of that "whole universe" is not an invariant. It is frame-dependent.

23. ### James RJust this guy, you know?Staff Member

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That's all I was saying. I'm glad we agree.

Sure. The "length" of any 4-vector is an invariant, and the tensor formulation of relativity is done in the way it is with the motivation, at one level, of avoiding having to talk about specific coordinate systems (i.e. frames of reference) at all. I always say that "relativity" is a bad name for the theory, since the real focus is more often "invariance".